Defining a Contact Form Locally. Obstructions?

[mus...@att.ne]

Hi,

Let M be a 3-manifold with a contact structure C , i.e., a nowhere-integrable hyperplane distribution.

I am trying to show that we can define locally (in a neighborhood Wx of each point x in M) a

form w , whose kernel is/defines the contact structure.

So, the idea is to define a 1-form whose kernel is precisely the hyperplane distribution.

My idea: for each x , we select first a basis B= {v1,v2} for the plane/hyperplane defined at x. We then

extend the basis B into a basis B' ={v1,v2,v3} for the tangent space at x, and we declare the form

w to satisfy w(v1)=w(v2)=0, and w(v3)=1 (every subspace is the kernel of linear map ). Now,

I don't see what the obstruction is to defining a global contact structure. Any ideas?

Thanks.

[mus...@att.ne]

Note that I am NOT musatov; I just like his e-mail, and I'm using my free speech , just like

s/he is.